Grade: 7 Unit: 3 Week: 2 Dates: 11/19 – 11/20 (2 Days) Content: Proportional Relationships, Tables, and Equations Theme Essential Question: How can you analyze proportional relationships and use them to solve real-world and mathematical problems?
Essential Questions:
What are the similarities and differences between the concepts of ratios, rates, and proportions?
How do rate tables expand the usage of the concepts of ratios, rates, and proportions?
What equation relationship is found in a proportional rate table? What is the connection between the table and equation?
How are ratios, rates, and proportions utilized in solutions to real-world and mathematical problems?
Standards:
7.RP.2a Decide whether two quantities are in a proportional relationship, e.g.. by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.
7.RP.2b Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.
7.RP.2c Represent proportional relationships by equations For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn..
7.RP.3 Analyze proportional relationships and use them to solve real-world and mathematical problems. Use proportional relationships to solve multistep ratio and percent problems.
Objectives:
The student will explore and discover that for every proportional relationship the ratios in the rate table are constant and this constant is known as the constant of proportionality. The student will distinguish between tables that are proportional/non-proportional.
The student will evaluate tables by observing the x (input) and y (output) to determine if the data is linear, nonlinear, or has no relationship. If linear the student will determine if it is an increase/decrease relationship.
The student will derive the equation for a proportional relationship, y = ax, where a is the constant of proportionality. The student will distinguish between equations that are proportional/non-proportional.
The student will identify whether a relationship presented in a table is proportional.
Reflections and/or Comments from your PCSSD 7th Grade Curriculum Team: As we provided students with the opportunity to look for the constant of proportionality and develop the equation relationship, y = ax, re-introduce Mathematical Practices # 7. Students will look for and make use of structure as they use tables to draw conclusions about proportional relationships. They will also look at tables and equations to determine if the relationship of one quantity to another is constant. They will find the relational equation y = ax for table that are proportional.
Due to the short week, it will be important to blend lesson 3-2 and 3-3. Additional time is also found in lesson 3-6. Please reference the Teacher Matrix for needed modifications.
Background Information Recommended: For a quick overview of the standard(s) to be addressed in this lesson, see Arizona’s Content Standards Reference Materials at **http://www.azed.gov/educator-certification/**
Assessment: Product
Continuation of unit portfolio
Entry #1
Each student is to
Find the constant of proportionality for the table.
Write the equation for the table. Identify the variables x and y.
Using the equation,
Given time find distance
Find the distance for times 52 sec.and 128 sec.
Given distance find time
Find the time for 204 in. and 1080 in.
Use the same strategies for the additional entries found in the student’s portfolio. Key Questions
How do you know, if the values in a table produces a proportional relationship?
If the table produces a proportional relationship, how do you find the constant of proportionality?
How does the constant of proportionality integrate into the equation?
How do you use the equations to find additional values, given input/output values?
What conclusion can you draw about a proportional relationship by looking at an equation and/or table?
Observable Student Behaviors (Performance)
Given a table, the student can determine if it is proportional.
Given a proportional rate table, the student can find the equation.
The student can use the equation to find additional values.
The student can explain the connections between the table and equation.
Mathematical Practices 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning.
Vocabulary:
Math
unit rate ratio common ratio proportional relationship constant constant of proportionality directly proportionality direct variation graphs tables equations
Suggested Activities:
Houghton Mifflin On Core Mathematics Middle School Grade 7 Unit 2-2, p. 39-42
ABC Mastering the Common Core in Mathematics
Proportional Relationships, Chapter 7.1, p. 73
Finding Relationships in Tables, Chapter 7.2, p 74
Finding Relationships in Equations, Chapter 7.4, p. 76
Gizmo Correlation
7.RP.2a
Direct Variation
Adjust the constant of variation and explore how the graph of the direct variation function changes in response.
Proportions and Common Multipliers
Complete a proportion using a graphical model. Use counters to fill cells in the numerators and denominators given. Use the visual pattern to determine how many counters to put in the missing numerator or denominator.
7.RP.2b
Beam to Moon (Ratios and Proportions)
Apply ratios and proportions to find the weight of a person on the moon (or on another planet). Weigh an object on Earth and on the moon and weigh the person on Earth. Then set up and solve the proportion of the Earth weights to the moon weights.
Dilations
Dilate a figure and investigate its resized image. See how scaling a figure affects the coordinates of its vertices, both in (x, y) form and in matrix form.
Perimeters and Areas of Similar Figures
Manipulate two similar figures and vary the scale factor to see what changes are possible under similarity. Explore how the perimeters and areas of two similar figures compare
Similar Figures - Activity B
Vary the scale factor and rotation of an image and compare it to the pre‑image. Determine how the angle measures and side lengths of the two figures are related.
Weight and Mass
Use a balance to measure mass and a spring scale to measure the weight of objects. Compare the masses and weights of objects on Earth, Mars, Jupiter, and the Moon.
7.RP.2c
Beam to Moon (Ratios and Proportions)
Apply ratios and proportions to find the weight of a person on the moon (or on another planet). Weigh an object on Earth and on the moon and weigh the person on Earth. Then set up and solve the proportion of the Earth weights to the moon weights.
Determining a Spring Constant
Place a pan on the end of a hanging spring. Measure how much the spring stretches when various masses are added to the pan. Create a graph of displacement vs. mass to determine the spring constant of the spring.
Estimating Population Size
Adjust the number of fish in a lake to be tagged and the number of fish to be recaptured. Use the number of tagged fish in the catch to estimate the number of fish in the lake.
Geometric Probability - Activity A
Randomly throw darts at a target and see what percent are "hits." Vary the size of the target and repeat the experiment. Study the relationship between the area of the target and the percent of darts that strike it
Theoretical and Experimental Probability
Experiment with spinners and compare the experimental probability of a particular outcome to the theoretical probability. Select the number of spinners, the number of sections on a spinner, and a favorable outcome of a spin. Then tally the number of favorable outcomes.
Highly Recommended (Problems are found in Lesson 3-3)
The Illustrative Mathematics Project offers guidance to states, assessment consortia, testing companies, and curriculum developers by illustrating the range and types of mathematical work that students will experience in a faithful implementation of the Common Core State Standards. The website features a clickable version of the Common Core in mathematics and the first round of "illustrations" of specific standards with associated classroom tasks and solutions.
JBHM 7th, GP 3, p.75-104
JBHM 8th, GP 1, p.207-220
Glencoe 7th Grade Mathematics Application and Concepts Course 2,
Chapter 7.2b, p. 296
Chapter 7.3, p, 297-301
Glencoe 7th Grade PreAlgebra, Chapter 6.3, p. 270-275
Beanstalk: The Measure of A Giant (A Math Adventure) Ann McCallum This fractured fairy tale tries to squeeze in a math lesson about ratios, but it isn’t successful. Jack wakes up to discover a beanstalk outside his window, climbs up, and befriends a lonely giant boy at the top. The two go off to play but quickly realize that the difference in their size is going to make most games difficult. Trying to play hoop ball, for example, proves to be impossible until Jack realizes that I need a hoop that as high for me as your hoop is for you. A few measurements later, the boys realize that Rays hoop is three times his height, so they figure out how tall Jacks should be, and fashion one for him. Once home, Jack decides to make a checker set for Ray and figures out what size to make it. The story ends with the two friends eating lunch outside with Jacks mother, who wishes for a word to describe the relationship between the sizes of two things, since the boys dealt with their differences so wonderfully. Jack decides that they should call it a Ray show, since Ray showed Jack that their things were the perfect size for each of them. The author sums the story up by explaining that today it is spelled ratio.
Mathematics in Children's Literature: Many children's books include math concepts and can be used to help teach them in a fun way. This website includes several annotated Lists of Children's Literature including the math concepts and grade levels.
The teaching Channel currently offers videos of K-12 mathematics teaching aligned with the Common Core Sate Standards, which would be perfect for professional development with teacher teams.
Interactivate is a set of free, online courseware for exploration in science and mathematics. It is comprised of activities, lessons, and discussions. The site is structured around collections of activities, lessons, and discussions.
Grade: 7 Unit: 3 Week: 2 Dates: 11/19 – 11/20 (2 Days)
Content: Proportional Relationships, Tables, and Equations
Theme Essential Question:
How can you analyze proportional relationships and use them to solve real-world and mathematical problems?
Essential Questions:
Standards:
Objectives:
Reflections and/or Comments from your PCSSD 7th Grade Curriculum Team:
As we provided students with the opportunity to look for the constant of proportionality and develop the equation relationship, y = ax, re-introduce Mathematical Practices # 7. Students will look for and make use of structure as they use tables to draw conclusions about proportional relationships. They will also look at tables and equations to determine if the relationship of one quantity to another is constant. They will find the relational equation y = ax for table that are proportional.
Due to the short week, it will be important to blend lesson 3-2 and 3-3. Additional time is also found in lesson 3-6. Please reference the Teacher Matrix for needed modifications.
Background Information
Recommended: For a quick overview of the standard(s) to be addressed in this lesson, see Arizona’s Content Standards Reference Materials at **http://www.azed.gov/educator-certification/**
Assessment:
Product
Use the same strategies for the additional entries found in the student’s portfolio.
Key Questions
Observable Student Behaviors (Performance)
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
tables equations
- Estimating Population Size
- Adjust the number of fish in a lake to be tagged and the number of fish to be recaptured. Use the number of tagged fish in the catch to estimate the number of fish in the lake.
- Geometric Probability - Activity A
- Randomly throw darts at a target and see what percent are "hits." Vary the size of the target and repeat the experiment. Study the relationship between the area of the target and the percent of darts that strike it
- Theoretical and Experimental Probability
- Experiment with spinners and compare the experimental probability of a particular outcome to the theoretical probability. Select the number of spinners, the number of sections on a spinner, and a favorable outcome of a spin. Then tally the number of favorable outcomes.
- Highly Recommended (Problems are found in Lesson 3-3)
The Illustrative Mathematics Project offers guidance to states, assessment consortia, testing companies, and curriculum developers by illustrating the range and types of mathematical work that students will experience in a faithful implementation of the Common Core State Standards. The website features a clickable version of the Common Core in mathematics and the first round of "illustrations" of specific standards with associated classroom tasks and solutions.Diverse Learners
- Odyssey (teacher discretion)
- Skill Tutor (teacher discretion)
- Math’scool: Unit A Mod 7.4
Homework: (Teacher Discretion)Terminology for Teachers:
Ethnicity/Culture | Immigration/Migration | Intercultural Competence | Socialization | Racism/Discrimination
High Yield Strategies
Similarities/Differences | Summarizing/Notetaking | Reinforcing/Recognition | Homework/Practice |
Non-Linguistic representation | Cooperative Learning | Objectives/Feedback |
Generating-Testing Hypothesis | Cues, Questions, Organizers
Resources:
Professional Texts:
Literary Texts
- Ratios
http://sci.tamucc.edu/~eyoung/ratio_prop_literature.html- Beanstalk: The Measure of A Giant (A Math Adventure) Ann McCallum This fractured fairy tale tries to squeeze in a math lesson about ratios, but it isn’t successful. Jack wakes up to discover a beanstalk outside his window, climbs up, and befriends a lonely giant boy at the top. The two go off to play but quickly realize that the difference in their size is going to make most games difficult. Trying to play hoop ball, for example, proves to be impossible until Jack realizes that I need a hoop that as high for me as your hoop is for you. A few measurements later, the boys realize that Rays hoop is three times his height, so they figure out how tall Jacks should be, and fashion one for him. Once home, Jack decides to make a checker set for Ray and figures out what size to make it. The story ends with the two friends eating lunch outside with Jacks mother, who wishes for a word to describe the relationship between the sizes of two things, since the boys dealt with their differences so wonderfully. Jack decides that they should call it a Ray show, since Ray showed Jack that their things were the perfect size for each of them. The author sums the story up by explaining that today it is spelled ratio.
- Mathematics in Children's Literature:
Click on the following link, http://libguides.nl.edu/mathinchildrenslit, and then look under Math and Literature Bibliographies.Many children's books include math concepts and can be used to help teach them in a fun way. This website includes several annotated Lists of Children's Literature including the math concepts and grade levels.
- Middle & High School: Literature in Mathematics
Many books include websites with lesson ideas.http://sci.tamucc.edu/~eyoung/middle_school_literature.html
- Lesson Plans for Using Literature in Middle and High School Mathematics (developed by Leonor and edited by Elaine)
http://sci.tamucc.edu/~eyoung/Literature%20webpages/Leonor/index.html- Miscellaneous Math and Children's Literature
http://sci.tamucc.edu/~eyoung/literature.htmlInformational Texts
Art, Music, and Media
Manipulatives
Games
https://www.teachingchannel.org/videos/junior-high-math-lesson
SMART Board Notebook file for Proportions Trail (Notebook
Videos
SMART Board Lessons, Promethean Lessons
Other Activities, etc.
- http://www.mathgoodies.com/lessons/vol5/division.html
- http://www.math.com/school/subject1/lessons/S1U1L12GL.html
- Interactivate is a set of free, online courseware for exploration in science and mathematics. It is comprised of activities, lessons, and discussions. The site is structured around collections of activities, lessons, and discussions.
http://www.shodor.org/interactivate/guide/- Activities and tools:
http://www.shodor.org/interactivate/activities/Language
Arts
Week 1
Week 2
Week 3
Week 4
Week 5
Week 6
Week 7
Week 8
7 Matrix
Accelerated 7
Matrix
Week 1
Week 2
Week 3
Week 4
Week 5
Week 6
Week 7
Week 8
Home K-2
Home 3-6
Home 6-8
Unit 1
Unit 2
Unit 3
Unit 4
Unit 5
Unit 6